# American Institute of Mathematical Sciences

March  2016, 5(1): 1-36. doi: 10.3934/eect.2016.5.1

## Well productivity index for compressible fluids and gases

 1 Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042 2 SUNY New Paltz, Department of Mathematics, 1 Hawk Dr, New Paltz, NY 12561, United States, United States

Received  September 2015 Revised  December 2015 Published  March 2016

We discuss the notion of the well productivity index (PI) for the generalized Forchheimer flow of fluid through porous media. The PI characterizes the well capacity with respect to drainage area of the well and in general is time dependent. In case of the slightly compressible fluid the PI stabilizes in time to the specific value, determined by the so-called pseudo steady state solution, [5,3,4]. Here we generalize our results from [4] in case of arbitrary order of the nonlinearity of the flow. In case of the compressible gas flow the mathematical model of the PI is studied for the first time. In contrast to slightly compressible fluid the PI stays almost'' constant for a long period of time, but then it blows up as time approaches the certain critical value. This value depends on the initial data (initial reserves) of the reservoir. The greater'' are the initial reserves, the larger is this critical value. We present numerical and theoretical results for the time asymptotic of the PI and its stability with respect to the initial data.
Citation: Eugenio Aulisa, Lidia Bloshanskaya, Akif Ibragimov. Well productivity index for compressible fluids and gases. Evolution Equations and Control Theory, 2016, 5 (1) : 1-36. doi: 10.3934/eect.2016.5.1
##### References:
 [1] D. G. Aronson, The porous medium equation, Lecture Notes in Mathematics, Springer, 1224 (1986), 1-46. doi: 10.1007/BFb0072687. [2] E. Aulisa, L. Bloshanskaya, L. Hoang and A. Ibragimov, Analysis of generalized Forchheimer flows of compressible fluids in porous media, J. Math. Phys., 50 (2009), 103102, 44pp. doi: 10.1063/1.3204977. [3] E. Aulisa, L. Bloshanskaya and A. Ibragimov, Long-term dynamics for well productivity index for nonlinear flows in porous media, J. Math. Phys., 52 (2011), 023506, 26pp. doi: 10.1063/1.3536463. [4] E. Aulisa, L. Bloshanskaya and A. Ibragimov, Time asymptotics of non-Darcy flows controlled by total flux on the boundary, J. Math. Sci., 184 (2012), 399-430. doi: 10.1007/s10958-012-0875-3. [5] E. Aulisa, A. Ibragimov, P. Valko and J. R. Walton, Mathematical framework of the well productivity index for fast Forchheimer (non-Darcy) flows in porous media, Mathematical Models and Methods in Applied Sciences, 19 (2009), 1241-1275. doi: 10.1142/S0218202509003772. [6] K. Aziz, L. Matta, S. Ko and G. S. Brar, Use of pressure, pressure-squared or pseudo-pressure in the analysis of transient pressure drawdown data from gas wells, Petroleum Society of Canada, 15 (1976), 9pp. doi: 10.2118/76-02-06. [7] J. Bear, Dynamics of Fluids in Porous Media, Dover Publications Inc., New York, 1988. doi: 10.1097/00010694-197508000-00022. [8] M. Bulíček, J. Málek and J. Žabenský, On generalized Stokes' and Brinkman's equations with a pressure- and shear-dependent viscosity and drag coefficient, Nonlinear Analysis: Real World Applications, 26 (2015), 109-132. doi: 10.1016/j.nonrwa.2015.05.004. [9] T. Christopher and O. Uche, Evaluating productivity index in a gas well using regression analysis, International Journal of Engineering Sciences&Research Technology, 3 (2014), 661-675. [10] L. P. Dake, Fundamentals of Reservoir Engineering, Elsevier, Amsterdam, 1983. [11] H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, 1856. [12] E. DiBenedetto, Degenerate Parabolic Equations, Springer, 1993. doi: 10.1007/978-1-4612-0895-2. [13] P. Forchheimer, Wasserbewegung durch boden zeit, Ver. Deut. Ing., 45 (1901), 1782. [14] L. Hoang, A. Ibragimov, T. Kieu and Z. Sobol, Stability of solutions to generalized Forchheimer equations of any degree, Journal of Mathematical Sciences, 210 (2015), 476-544. doi: 10.1007/s10958-015-2576-1. [15] A. Ibragimov, D. Khalmanova, P. P. Valko and J. R. Walton, On a mathematical model of the productivity index of a well from reservoir engineering, SIAM Journal on Applied Mathematics, 65 (2005), 1952-1980. doi: 10.1137/040607654. [16] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968. [17] E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, vol. 171 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1998, Translated from the 1971 Russian original by Tamara Rozhkovskaya, With a preface by Nina Ural'tseva. [18] D. Li and T. W. Engler, Literature review on correlations of the non-Darcy coefficient, SPE, (2001), 8pp. doi: 10.2118/70015-MS. [19] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites non Linéaires, Dunod, 1969. [20] M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, International Human Resources Development, 1982. doi: 10.1097/00010694-193808000-00008. [21] K. Nakshatrala and K. Rajagopal, A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium, International Journal for Numerical Methods in Fluids, 67 (2011), 342-368. doi: 10.1002/fld.2358. [22] L. E. Payne, J. C. Song and B. Straughan, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, The Royal Society, 455 (1999), 2173-2190. doi: 10.1098/rspa.1999.0398. [23] L. E. Payne and B. Straughan, Convergence and continuous dependence for the brinkman-forchheimer equations, Studies in Applied Mathematics, 102 (1999), 419-439. doi: 10.1111/1467-9590.00116. [24] C. A. Pereira, H. Kazemi and E. Ozkan, Combined effect of non-Darcy flow and formation damage on gas well performance of dual-porosity and dual-permeability reservoirs, SPEJ, 9 (2006), 543-552. doi: 10.2118/90623-PA. [25] R. Raghavan, Well Test Analysis, Prentice Hall, New York, 1993. [26] S. Srinivasan and K. Rajagopal, A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations, International Journal of Non-Linear Mechanics, 58 (2014), 162-166. doi: 10.1016/j.ijnonlinmec.2013.09.004. [27] J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007, Mathematical theory. [28] C. Wolfsteiner, L. Durlofsky and K. Aziz, Calculation of well index for nonconventional wells on arbitrary grids, Computational Geosciences, 7 (2003), 61-82.

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##### References:
 [1] D. G. Aronson, The porous medium equation, Lecture Notes in Mathematics, Springer, 1224 (1986), 1-46. doi: 10.1007/BFb0072687. [2] E. Aulisa, L. Bloshanskaya, L. Hoang and A. Ibragimov, Analysis of generalized Forchheimer flows of compressible fluids in porous media, J. Math. Phys., 50 (2009), 103102, 44pp. doi: 10.1063/1.3204977. [3] E. Aulisa, L. Bloshanskaya and A. Ibragimov, Long-term dynamics for well productivity index for nonlinear flows in porous media, J. Math. Phys., 52 (2011), 023506, 26pp. doi: 10.1063/1.3536463. [4] E. Aulisa, L. Bloshanskaya and A. Ibragimov, Time asymptotics of non-Darcy flows controlled by total flux on the boundary, J. Math. Sci., 184 (2012), 399-430. doi: 10.1007/s10958-012-0875-3. [5] E. Aulisa, A. Ibragimov, P. Valko and J. R. Walton, Mathematical framework of the well productivity index for fast Forchheimer (non-Darcy) flows in porous media, Mathematical Models and Methods in Applied Sciences, 19 (2009), 1241-1275. doi: 10.1142/S0218202509003772. [6] K. Aziz, L. Matta, S. Ko and G. S. Brar, Use of pressure, pressure-squared or pseudo-pressure in the analysis of transient pressure drawdown data from gas wells, Petroleum Society of Canada, 15 (1976), 9pp. doi: 10.2118/76-02-06. [7] J. Bear, Dynamics of Fluids in Porous Media, Dover Publications Inc., New York, 1988. doi: 10.1097/00010694-197508000-00022. [8] M. Bulíček, J. Málek and J. Žabenský, On generalized Stokes' and Brinkman's equations with a pressure- and shear-dependent viscosity and drag coefficient, Nonlinear Analysis: Real World Applications, 26 (2015), 109-132. doi: 10.1016/j.nonrwa.2015.05.004. [9] T. Christopher and O. Uche, Evaluating productivity index in a gas well using regression analysis, International Journal of Engineering Sciences&Research Technology, 3 (2014), 661-675. [10] L. P. Dake, Fundamentals of Reservoir Engineering, Elsevier, Amsterdam, 1983. [11] H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, 1856. [12] E. DiBenedetto, Degenerate Parabolic Equations, Springer, 1993. doi: 10.1007/978-1-4612-0895-2. [13] P. Forchheimer, Wasserbewegung durch boden zeit, Ver. Deut. Ing., 45 (1901), 1782. [14] L. Hoang, A. Ibragimov, T. Kieu and Z. Sobol, Stability of solutions to generalized Forchheimer equations of any degree, Journal of Mathematical Sciences, 210 (2015), 476-544. doi: 10.1007/s10958-015-2576-1. [15] A. Ibragimov, D. Khalmanova, P. P. Valko and J. R. Walton, On a mathematical model of the productivity index of a well from reservoir engineering, SIAM Journal on Applied Mathematics, 65 (2005), 1952-1980. doi: 10.1137/040607654. [16] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968. [17] E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, vol. 171 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1998, Translated from the 1971 Russian original by Tamara Rozhkovskaya, With a preface by Nina Ural'tseva. [18] D. Li and T. W. Engler, Literature review on correlations of the non-Darcy coefficient, SPE, (2001), 8pp. doi: 10.2118/70015-MS. [19] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites non Linéaires, Dunod, 1969. [20] M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, International Human Resources Development, 1982. doi: 10.1097/00010694-193808000-00008. [21] K. Nakshatrala and K. Rajagopal, A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium, International Journal for Numerical Methods in Fluids, 67 (2011), 342-368. doi: 10.1002/fld.2358. [22] L. E. Payne, J. C. Song and B. Straughan, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, The Royal Society, 455 (1999), 2173-2190. doi: 10.1098/rspa.1999.0398. [23] L. E. Payne and B. Straughan, Convergence and continuous dependence for the brinkman-forchheimer equations, Studies in Applied Mathematics, 102 (1999), 419-439. doi: 10.1111/1467-9590.00116. [24] C. A. Pereira, H. Kazemi and E. Ozkan, Combined effect of non-Darcy flow and formation damage on gas well performance of dual-porosity and dual-permeability reservoirs, SPEJ, 9 (2006), 543-552. doi: 10.2118/90623-PA. [25] R. Raghavan, Well Test Analysis, Prentice Hall, New York, 1993. [26] S. Srinivasan and K. Rajagopal, A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations, International Journal of Non-Linear Mechanics, 58 (2014), 162-166. doi: 10.1016/j.ijnonlinmec.2013.09.004. [27] J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, Oxford, 2007, Mathematical theory. [28] C. Wolfsteiner, L. Durlofsky and K. Aziz, Calculation of well index for nonconventional wells on arbitrary grids, Computational Geosciences, 7 (2003), 61-82.
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